Long-Range GL(n) Integrable Spin Chains and Plane-Wave Matrix Theory

TL;DR

This paper investigates the structure of long-range integrable spin chains with gl(n) symmetry, deriving their Hamiltonian and Bethe ansatz, and applies findings to plane-wave matrix theory, revealing non-integrability beyond first order.

Contribution

It derives the Hamiltonian and Bethe ansatz for the most general long-range gl(n) spin chain and applies these results to plane-wave matrix theory, showing its non-integrability beyond first order.

Findings

Derived the Hamiltonian and Bethe ansatz at four perturbative orders.

Proposed Bethe equations for all orders and identified moduli.

Found that the plane-wave matrix theory Hamiltonian is not integrable beyond first order.

Abstract

Quantum spin chains arise naturally from perturbative large-N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbor type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of gl(n). We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.